Computability in Timed Sets in Opetaa, Estonia Robin Cockett Joaqu´ ın D´ ıaz-Bo¨ ıls Jonathan Gallagher Pavel Hrubeˇ s University ofCalgary February 4, 2013
Motivation Explicit versus implicit Timed sets Where are we going? Timing maps Complexity orders Restriction categories Computability in timed sets Iteration ... Splitting idempotents Getting non-zero size .. Computability Powerful objects Program objects Turing structure 2 of 63
� � Explicit versus implicit EXPLICIT IMPLICIT (model) (axiomatic theory) PROBLEM: Need to know what one is modelling ... Need to know how to axiomatize the phenomenon 3 of 63
� � Explicit versus implicit EXPLICIT COMPUTABILITY (particular model) IMPLICIT COMPUTABILITY (axiomatic theory) 4 of 63
Explicit computability • Turing machine computing (partial) functions • Kleene’s first model (natural numbers are codes for machines which act on numbers) • Oracle computability (jump operators) • Combinatory and λ -algebras • Domain theory models. 5 of 63
Implicit computability • Axiomatic/logic approaches to computability ... • Combinatory logic and λ -calculus ... • Turing categories Turing categories = abstract computability MANY non-standard models!! ... all models are Turing categories. 6 of 63
� � Explicit versus implicit EXPLICIT COMPLEXITY (particular model) IMPLICIT COMPLEXITY (axiomatic theory) 7 of 63
Explicit complexity • Time complexity: counting the ticks of a Turing/computing machine • Space complexity: counting the storage required by a Turing/computing machine Want these notions to be independent of the machine model ... Are they? Well not really! e.g. Turing machine versus pointer models at low complexity 8 of 63
Implicit complexity Why do it? • Theoretical understanding of complexity ... wide variety of different models relationship between different models correspondence between axiomatic features and complexity • Type checking for complexity real-time applications ... 9 of 63
This talk looks at the explicit models complexity theorists themselves use!!! but with categorical eyes! Part of the program of abstract computability: unifies complexity and computability. 10 of 63
Functional Complexity in a Timed Maps “Universe” 11 of 63
Functional Complexity in a Timed Maps “Universe” A surprise connection between partiality and complexity A categorical model/semantics of basic complexity theory A construction that builds models of computability whose total maps are precisely the maps of a given functional complexity class: P -time, Log-space, and above ... I.e. mimic what complexity theorists do ... BUT with categorical eyes. 12 of 63
� � The timing of a partial map as a primitive Start with a notion of timing/costing a partial map: f A B f ( x ) ↓ ⇔ | x | f ↓ |·| f N • A partial function f may have different timings • Think of each timing as the cost (time/space/resource) of computing f by an algorithm 13 of 63
ASIDE: What is cost? We shall assume cost is a natural number BUT the theory works more generally! A size monoid is a partially ordered commutative monoid ( M , 0 , + , ≤ ) such that • 0 ≤ x for all x ∈ M , • x ≤ x ′ and y ≤ y ′ implies x + y ≤ x ′ + y ′ . Examples: N , R ≥ 0 , N × N ... In fact, given any commutative monoid A set x ≤ y if there is a z with x + z = y then x ∼ y ≡ x ≤ y & y ≤ x then size( A ) = A / ∼ is the universal size monoid associated with A . Note: size monoids are orthogonal to commutative groups. 14 of 63
� � � � � � The Category of Timed Sets TSet: • Objects: Sets • Maps: Timed partial functions • Identity: The identity function with 0 cost • Composition: g fg f A B C = A C |·| f |·| g |·| f + | f ( · ) | g N N N 15 of 63
The Category of Timed Sets Too restrictive ... Two maps are equal only if their timing are exactly the same ... Need to capture O ( ) the order of complexity ... 16 of 63
Complexity Orders An additive complexity order C is a class of monotone functions P : N − → N such that C is: • down-closed: P ∈ C and Q ≤ P then Q ∈ C ; • closed to composition: if P , Q in C then PQ ∈ C ; • additive: 0 ∈ C and if P , Q ∈ C then P + Q ∈ C . 17 of 63
Examples of complexity orders Linear: L = � λ x . nx | n ∈ N � Polynomial: � � n � a i x i | n ∈ N P = λ x . i =1 Where � � denotes down-closure. 18 of 63
C -ordering Every complexity order C induces a preorder enrichment on the maps of TSet, f ≤ C g : • g ( x ) ↓ implies f ( x ) ↓ and g ( x ) = f ( x ); • there is a P ∈ C such that for all x , | x | f ≤ P ( | x | g ). 19 of 63
C -equivalence C -equivalence is the congruence f = C g if: f ≤ C g and g ≤ C f E.g. f = L g if | x | f ≤ m | x | g and | x | g ≤ n | x | f 20 of 63
Partiality: Restriction Categories For each map f : A − → B : a restriction idempotent f : A − → A such that [ R . 1 ] f f = f [ R . 3 ] f g = f g [ R . 2 ] f g = g f [ R . 4 ] f h = fh f • A general framework for partiality [Cockett and Lack 2002] E.g. Sets and partial functions: f is domain of definition • P-categories [Robinson and Rosolini 1988] • Influential paper by Robert Di Paola and Alex Heller on “dominical categories” (1986) initiates abstract computability. 21 of 63
Totality in Restriction Category Recall that a map in a restriction category is total in case f = 1 . 22 of 63
Timed Sets and Restriction Structure For TSet, the desired restriction is ( f , |·| f ) = ( f , |·| f ). However, this is not a restriction structure since [R.1] fails: ( f , |·| f ) ( f , |·| f ) = ( f , |·| f + |·| f ) � = ( f , |·| f ) 23 of 63
The Restriction Category of Timed Sets TSet may be quotiented by the congruence = C . Proposition. For any complexity order C , TSet / C is a restriction category where ( f , |·| f ) = ( f , |·| f ) . 24 of 63
Linking Complexity Order and Partiality Every restriction category is partial order enriched by f ≤ g : f g = f . Lemma. In TSet / C , f ≤ g if and only if g ≤ C f . 25 of 63
Iteration in a restriction category f ⋆ g : f n g (for at most one n ) Intuitively g ⊔ fg ⊔ ffg ⊔ · · · 26 of 63
Iteration in a restriction category Iterate(f,g)(x) = while (x in dom(f)) x := f (x) g(x) Iteration: one way to obtain computability ... 27 of 63
Disjoint joins Proposition. For every C , TSet / C has disjoint joins. What does that mean? 28 of 63
Disjoint joins Disjointness, means that “domains” do not overlap f g = ∅ The join of disjoint maps f , g is the join, ⊔ , with respect to ≤ . Must also be ”stable” with respect to composition: h ( f ⊔ g ) = hf ⊔ hg 29 of 63
Disjoint joins and iteration Need disjoint joins for iteration ... � f n g f ⋆ g = g ⊔ fg ⊔ ffg ⊔ · · · = n Also need � � f n g = C f ′ n g ′ n n whenever f = C f ′ and g = C g ′ . This requires the complexity order satisfy an extra laxness condition ... 30 of 63
Distributive Restriction Categories Proposition. [Cockett and Lack 2007] For a restriction category Distributivity ⇒ Extensiveness ⇒ Disjoint Joins 31 of 63
The Distributive Restriction Category of Timed Sets Proposition. For every complexity order C , TSet / C is a distributive restriction cate- gory. 32 of 63
TSet / C has a restriction terminal object 1 = { ⋆ } is the restriction terminal object. ! A : A − → 1 is always defined and has zero cost. Thus, for any f : A − → 1 , f = f ! A 33 of 63
� � � � Restriction products The binary restriction product of A , B is A × B with total projections π 0 , π 1 and a unique pairing such that in U g f � f , g � � B A × B A π 0 π 1 � f , g � π 0 = g f and � f , g � π 1 = f g . 34 of 63
TSet / C has restriction products A × B is as in Sets. Projections, π , are always defined and have zero cost. � ( f , |·| f ) , ( g , |·| g ) � := ( � f , g � , |·| � f , g � ) where | x | � f , g � := | x | f + | x | g 35 of 63
TSet / C has an initial object 0 = ∅ is the initial object. Note also that TSet / C has nowhere defined maps: ∅ := ( ∅ , ∅ ) : A − → B 36 of 63
TSet / C has coproducts A + B is as in Sets. Coprojections σ are always defined and have zero cost. [( f , |·| f ) , ( g , |·| g )] = ([ f , g ] , |·| [ f , g ] ) where |·| [ f , g ] = [ |·| f , |·| g ] 37 of 63
TSet / C is distributive The map [ A + σ B , A + σ C ] : ( A × B ) + ( A × C ) − → A × ( B + C ) is an isomorphism in Sets, and is zero cost. 38 of 63
Iteration f : A − → A g : A − → B f , g disjoint f ⋆ g : A − → B where [Conway 1971]: For example W.1 : W.1 ( fg ) ⋆ h = h ⊔ f (( gf ) ⋆ ( gh )) f ⋆ h W.2 ( f ⊔ g ) ⋆ h = ( f ⋆ g ) ⋆ ( f ⋆ h ) = h ⊔ f ( f ⋆ h ) W.3 ( f ⋆ g ) h = f ⋆ ( gh ) = h ⊔ f ( h ⊔ f ( f ⋆ h )) W.4 1 × ( f ⋆ g ) = (1 × f ) ⋆ (1 × g ) = h ⊔ fh ⊔ f 2 ( f ⋆ h ) W.5 f ≤ f ′ , g ≤ g ′ then f ⋆ g ≤ f ′ ⋆ g ′ = · · · 39 of 63
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